Optimal Upper and Lower Bounds for Boolean Expressions by Dissociation

This paper develops upper and lower bounds for the probability of Boolean expressions by treating multiple occurrences of variables as independent and assigning them new individual probabilities. Our technique generalizes and extends the underlying idea of a number of recent approaches which are varyingly called node splitting, variable renaming, variable splitting, or dissociation for probabilistic databases. We prove that the probabilities we assign to new variables are the best possible in some sense.

💡 Research Summary

The paper tackles the notoriously hard problem of computing the exact probability of Boolean expressions, a task that is #P‑complete in general. To obtain tractable approximations, the authors introduce a unifying technique called dissociation. The core idea is to treat each occurrence of a variable that appears multiple times in a Boolean formula as an independent copy, assign each copy its own probability, and then evaluate the probability of the resulting “dissociated” expression. By carefully choosing the probabilities of the new variables, the dissociated expression yields either an upper or a lower bound on the original probability.

The authors first set up notation: a set of independent Boolean random variables (x = (x_1,\dots,x_k)) with primitive event probabilities (p_i = P